Generalization of Jacobi’s theorem on the last multiplier

To satisfy the conditions of Jacobi’s theorem on the last multiplier, the existence of an invariant measure and a sufficient number of independent first integrals are needed. In this case, the system can be locally integrated by quadratures. There are examples of systems for which the existence of partial first integrals is sufficient for the possibility of integration by quadratures. Moreover, integration by quadratures occurs at the level of partial first integrals. In this paper, Jacobi’s theorem on the last multiplier is extended to the general situation when the first integrals include partial ones.